Chapter 2: Logarithms

Exercise 2.1 Solved Notes

A comprehensive, step-by-step solved reference guide for converting numbers between Scientific Notation and Ordinary (Standard) Notation, completely cleaned of all watermarks and web brandings.

Core Mathematical Concept Theory

Scientific Notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. A number is written in scientific notation when it is expressed in the form:

a × 10n

Where:

  • a is a decimal number (coefficient) such that 1 ≤ |a| < 10 (i.e. exactly one non-zero digit before the decimal point).
  • n is an integer (exponent), representing the number of places the decimal point was shifted.

Rule of Decimal Movement:

  • Move decimal point to the LEFT → Exponent is POSITIVE (+n).
  • Move decimal point to the RIGHT → Exponent is NEGATIVE (-n).
Question 1 Scientific Notation
Express the following numbers in scientific notation.
(i) 2,000,000

Solution:

Place the decimal after the first non-zero digit: 2.000000

We move the decimal point 6 places to the left.

2 0 0 0 0 0 0 .
2,000,000 = 2.0 × 106
(ii) 48,900

Solution:

Place the decimal after the first non-zero digit: 4.8900

We move the decimal point 4 places to the left.

4 8 9 0 0 .
48,900 = 4.89 × 104
(iii) 0.0042

Solution:

Place the decimal after the first non-zero digit (4): 4.2

We move the decimal point 3 places to the right.

. 0 0 4 2
0.0042 = 4.2 × 10-3
(iv) 0.0000009

Solution:

Place the decimal after the first non-zero digit (9): 9.0

We move the decimal point 7 places to the right.

. 0 0 0 0 0 0 9
0.0000009 = 9.0 × 10-7
(v) 73 × 103

Solution:

First, write the coefficient 73 in scientific notation: 73 = 7.3 × 101 (decimal point shifted 1 place left).

Substitute this back into the original expression and add the exponents:

73 × 103 = (7.3 × 101) × 103
= 7.3 × 101 + 3
= 7.3 × 104
(vi) 0.65 × 102

Solution:

First, write the coefficient 0.65 in scientific notation: 0.65 = 6.5 × 10-1 (decimal point shifted 1 place right).

Substitute this back into the original expression and add the exponents:

0.65 × 102 = (6.5 × 10-1) × 102
= 6.5 × 10-1 + 2
= 6.5 × 101
Question 2 Ordinary Notation
Express the following numbers in ordinary notation.
(i) 8.04 × 102

Solution:

Since the exponent is positive 2, we move the decimal point 2 places to the right:

8.04 × 102 = 8.04 × 100 = 804
(ii) 3 × 105

Solution:

Since the exponent is positive 5, we move the decimal point 5 places to the right (filling in zeros):

3 × 105 = 3.00000 × 105 = 300,000
(iii) 1.5 × 10-2

Solution:

Since the exponent is negative -2, we move the decimal point 2 places to the left:

1.5 × 10-2 = 0.015
(iv) 1.77 × 107

Solution:

Since the exponent is positive 7, we move the decimal point 7 places to the right (adding five zeros after the 7s):

1.77 × 107 = 17,700,000
(v) 5.5 × 10-6

Solution:

Since the exponent is negative -6, we move the decimal point 6 places to the left (adding five placeholder zeros before the first 5):

5.5 × 10-6 = 0.0000055
(vi) 4 × 10-5

Solution:

Since the exponent is negative -5, we move the decimal point 5 places to the left:

4 × 10-5 = 0.00004
Question 3 Word Problem
The speed of light is approximately 3 × 108 meters per second. Express it in standard form.
Solution

Given speed of light in scientific notation: 3 × 108 m/s

To convert to standard (ordinary) form, since the exponent is 8 (positive), we move the decimal point 8 places to the right:

3 × 108 = 300,000,000\text{ meters per second}

Answer: The speed of light is 300,000,000 m/s (or 300 million meters per second).

Question 4 Word Problem
The circumference of the Earth at the equator is about 40,075,000 metres. Express this number in scientific notation.
Solution

Given equator circumference: 40,075,000 m

To express in scientific notation, we shift the decimal point to sit right after the first non-zero digit (4): 4.0075000

The decimal point is shifted 7 places to the left, which gives a positive exponent of 7:

40,075,000 = 4.0075 × 107\text{ meters}

Answer: The equator circumference is 4.0075 × 107 m.

Question 5 Word Problem
The diameter of Mars is 6.779 × 103 km. Express this number in standard form.
Solution

Given diameter of Mars: 6.779 × 103 km

To convert to standard (ordinary) form, since the exponent is 3 (positive), we move the decimal point 3 places to the right:

6.779 × 103 = 6779\text{ km}

Answer: The diameter of Mars is 6,779 km.

Question 6 Word Problem
The diameter of Earth is about 1.2756 × 104 km. Express this number in standard form.
Solution

Given diameter of Earth: 1.2756 × 104 km

To convert to standard (ordinary) form, since the exponent is 4 (positive), we move the decimal point 4 places to the right:

1.2756 × 104 = 12756\text{ km}

Answer: The diameter of Earth is 12,756 km.

Interactive Utility Sandbox Interactive

Convert any decimal number into scientific notation or vice versa. Try typing values like 150000 or 0.00045.

Decimal to Scientific

Scientific to Decimal

Test your understanding of scientific notations with this interactive quiz. Answer all 5 questions to receive your score!

1. What is 0.00045 expressed in scientific notation?
(a) 4.5 × 104
(b) 4.5 × 10-4   (Correct)
(c) 0.45 × 10-3
(d) 45.0 × 10-5
Correct Answer: (b) 4.5 × 10-4
Reason: The decimal point is shifted 4 places to the right, which gives a negative exponent of -4.
2. Express 3.05 × 105 in ordinary decimal form.
(a) 30,500
(b) 3,050,000
(c) 305,000   (Correct)
(d) 0.0000305
Correct Answer: (c) 305,000
Reason: Since the exponent is +5, shift the decimal point 5 places to the right: 3.05 → 30.5 → 305 → 3050 → 30500 → 305000.
3. Convert 82 × 10-3 to standard scientific notation form.
(a) 8.2 × 10-2   (Correct)
(b) 8.2 × 10-4
(c) 0.82 × 10-2
(d) 820 × 10-4
Correct Answer: (a) 8.2 × 10-2
Reason: First rewrite 82 as 8.2 × 101. Then combine with 10-3: 101 + (-3) = 10-2.
4. The diameter of a typical red blood cell is about 0.000007 meters. In scientific notation, this is:
(a) 7.0 × 106 m
(b) 7.0 × 10-6 m   (Correct)
(c) 7.0 × 10-5 m
(d) 0.7 × 10-6 m
Correct Answer: (b) 7.0 × 10-6 m
Reason: Shift the decimal point 6 places to the right to write 7.0, so the exponent is -6.
5. If a number is written in standard scientific form as a × 10n, what are the limits on coefficient 'a'?
(a) 0 ≤ a < 10
(b) 1 ≤ a < 10   (Correct)
(c) 1 ≤ a ≤ 10
(d) No limit
Correct Answer: (b) 1 ≤ a < 10
Reason: The coefficient must have exactly one non-zero digit before the decimal point, which means it must be at least 1 and strictly less than 10.